Answer :
To determine the values of [tex]\( m \)[/tex] for which the graph of the quadratic equation [tex]\( y = 3x^2 + 7x + m \)[/tex] has two [tex]\( x \)[/tex]-intercepts, we need to consider the quadratic formula and the concept of the discriminant.
A quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] has two distinct real [tex]\( x \)[/tex]-intercepts when the discriminant, given by [tex]\( b^2 - 4ac \)[/tex], is positive.
Let's break it down step-by-step:
1. Identify the coefficients:
In the equation [tex]\( y = 3x^2 + 7x + m \)[/tex], we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
2. Calculate the discriminant:
Using the formula for the discriminant [tex]\( b^2 - 4ac \)[/tex], we plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Discriminant} = 7^2 - 4 \times 3 \times m
\][/tex]
3. Simplify the discriminant:
Calculate the known parts:
[tex]\[
49 - 12m
\][/tex]
4. Set the discriminant greater than 0:
For the quadratic to have two distinct [tex]\( x \)[/tex]-intercepts, we need:
[tex]\[
49 - 12m > 0
\][/tex]
5. Solve the inequality for [tex]\( m \)[/tex]:
Rearrange the inequality:
[tex]\[
49 > 12m
\][/tex]
6. Isolate [tex]\( m \)[/tex]:
Divide both sides by 12:
[tex]\[
m < \frac{49}{12}
\][/tex]
Thus, the graph of [tex]\( y = 3x^2 + 7x + m \)[/tex] will have two [tex]\( x \)[/tex]-intercepts for values of [tex]\( m \)[/tex] that satisfy [tex]\( m < \frac{49}{12} \)[/tex].
A quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] has two distinct real [tex]\( x \)[/tex]-intercepts when the discriminant, given by [tex]\( b^2 - 4ac \)[/tex], is positive.
Let's break it down step-by-step:
1. Identify the coefficients:
In the equation [tex]\( y = 3x^2 + 7x + m \)[/tex], we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
2. Calculate the discriminant:
Using the formula for the discriminant [tex]\( b^2 - 4ac \)[/tex], we plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Discriminant} = 7^2 - 4 \times 3 \times m
\][/tex]
3. Simplify the discriminant:
Calculate the known parts:
[tex]\[
49 - 12m
\][/tex]
4. Set the discriminant greater than 0:
For the quadratic to have two distinct [tex]\( x \)[/tex]-intercepts, we need:
[tex]\[
49 - 12m > 0
\][/tex]
5. Solve the inequality for [tex]\( m \)[/tex]:
Rearrange the inequality:
[tex]\[
49 > 12m
\][/tex]
6. Isolate [tex]\( m \)[/tex]:
Divide both sides by 12:
[tex]\[
m < \frac{49}{12}
\][/tex]
Thus, the graph of [tex]\( y = 3x^2 + 7x + m \)[/tex] will have two [tex]\( x \)[/tex]-intercepts for values of [tex]\( m \)[/tex] that satisfy [tex]\( m < \frac{49}{12} \)[/tex].
